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Statistics / Probability and Statistical Inference 9 / Chapter 2.6 / Problem 13E

Probability and Statistical Inference | 9th Edition | ISBN: 9780321923271 | Authors: Robert V. Hogg, Elliot Tanis, Dale Zimmerman

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Assume that a policyholder is four times more likely to

Chapter 2, Problem 13E

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QUESTION:

Problem 13E

Assume that a policyholder is four times more likely to file exactly two claims as to file exactly three claims. Assume also that the number X of claims of this policyholder is Poisson. Determine the expectation E(X2).

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QUESTION:

Problem 13E

Assume that a policyholder is four times more likely to file exactly two claims as to file exactly three claims. Assume also that the number X of claims of this policyholder is Poisson. Determine the expectation E(X2).

ANSWER:

Answer :

Step 1 of 3 :

We have to assume that a policyholder is four times more likely to file exactly two claims as to file exactly three times. And also that the number X of claims of this policyholder is poisson.

The claim is to determine the E( ${X}^{2}$ )

We have, P(x = 2) = 4P(x = 3)

X~poisson distribution

P(X=x) = $\frac{{\lambda{}}^{x}{e}^{-\lambda{}}}{x!}$

From the equation P(x = 2) = 4P(x = 3)

We get,

$\frac{{\lambda{}}^{2}{e}^{-\lambda{}}}{2!}$ = $\frac{{\lambda{}}^{3}{e}^{-\lambda{}}}{3!}$

$\frac{{e}^{-\lambda{}}{e}^{\lambda{}}}{2!}$ = $\frac{{\lambda{}}^{3}{\lambda{}}^{-2}}{3!}$

$\frac{1}{2}$ = $\frac{{\lambda{}}^{\ }}{6}$

${\lambda{}}^{\ }$ = 3

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